Krackhardt Hierarchy Measures
Also known as: Krackhardt GTD, graph-theoretic dimensions of hierarchy, Krackhardt connectedness-hierarchy-efficiency-LUB, out-tree hierarchy measures
Krackhardt's graph-theoretic dimensions provide four indices that together measure how closely a directed network approximates a pure hierarchy — formally, an out-tree. The dimensions are connectedness (is everyone linked?), hierarchy (are ties asymmetric, i.e., non-reciprocated?), efficiency (are there no redundant ties?), and least upper bound (does every pair share a common superior?). Each is scaled from 0 to 1, and a network scoring 1 on all four is a perfect hierarchy.
Key highlights
- Decomposes 'hierarchy' into four distinct, separately interpretable structural properties.
- Each dimension is a normalized 0–1 index, easing comparison across networks.
- Grounded in a precise graph-theoretic ideal (the out-tree) rather than an ad hoc notion.
- Diagnoses the specific way a structure deviates from a pure hierarchy.
Intuition
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How it works
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When to use it
Use Krackhardt's dimensions when you want to quantify how hierarchical a directed network is and diagnose precisely how it departs from a pure hierarchy — useful for comparing organizational authority structures, reporting networks, or any directed relation expected to be tree-like. Each dimension isolates a distinct property, so the profile is more informative than a single index. The measures are defined on the reachability (transitive closure) of a directed graph and presume the out-tree is the relevant ideal; they are descriptive of structure, not of process or outcomes, and (as Everett and Krackhardt later noted) some original formulations needed refinement for edge cases. They do not apply to undirected networks.
Strengths & limitations
- Decomposes 'hierarchy' into four distinct, separately interpretable structural properties.
- Each dimension is a normalized 0–1 index, easing comparison across networks.
- Grounded in a precise graph-theoretic ideal (the out-tree) rather than an ad hoc notion.
- Diagnoses the specific way a structure deviates from a pure hierarchy.
- Defined for directed networks via reachability; not applicable to undirected relations.
- Presumes the out-tree is the relevant ideal of hierarchy, which may not fit all authority forms.
- Some original index formulations have edge-case issues later refined in the literature.
- Descriptive of structure only; says nothing about how the hierarchy functions or performs.
Common pitfalls
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Applications
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Frequently asked
What are the four dimensions and what does each capture?
Connectedness: whether all actors form one connected structure. Hierarchy: whether reachability is asymmetric, i.e., no reciprocated chains of authority. Efficiency: whether the structure avoids redundant ties beyond the minimum needed. Least upper bound: whether every pair of actors shares a common superior. A network scoring 1 on all four is a perfect out-tree hierarchy.
Why use four separate measures instead of one?
Because a network can be hierarchical in some respects and not others — for example, fully connected and acyclic (high connectedness and hierarchy) but with redundant ties (low efficiency) or without a unified apex (low LUB). The four dimensions are logically independent, so reporting them separately diagnoses exactly how and where a structure departs from a pure hierarchy.
On what relation are the dimensions computed?
They are defined on the reachability relation — the transitive closure of the directed network — rather than the raw ties. Two actors are 'related' if one can reach the other through a directed path. Computing the measures on the adjacency matrix instead of its transitive closure misstates connectedness, hierarchy, and the least-upper-bound property.
Sources
- 1.Krackhardt, D. (1994). Graph theoretical dimensions of informal organizations. In K. M. Carley & M. J. Prietula (Eds.), Computational Organization Theory (pp. 89–111). Lawrence Erlbaum Associates.
- 2.Everett, M. G., & Krackhardt, D. (2012). A second look at Krackhardt's graph theoretical dimensions of informal organizations. Social Networks, 34(2), 159–163.
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ScholarGate. (2026, June 22). Krackhardt Hierarchy Measures. ScholarGate. https://scholargate.app/sociology/krackhardt-hierarchy